arXiv subjects

Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 379 records · Page 21Linked to original sources

arXiv2015

$\ell_1$-regularized Neural Networks are Improperly Learnable in Polynomial Time

We study the improper learning of multi-layer neural networks. Suppose that the neural network to be learned has $k$ hidden layers and that the $\ell_1$-norm of the incoming weights of any neuron is bounded by $L$. We present a kernel-based method, such that with probability at least $1 - δ$, it learns a predictor whose generalization error is at most $ε$ worse than that of the neural network. The sample complexity and the time complexity of the presented method are polynomial in the input dimension and in $(1/ε,\log(1/δ),F(k,L))$, where $F(k,L)$ is a function depending on $(k,L)$ and on the activation function, independent of the number of neurons. The algorithm applies to both sigmoid-like activation functions and ReLU-like activation functions. It implies that any sufficiently sparse neural network is learnable in polynomial time.

cs.LG↗

arXiv2015

Splash: User-friendly Programming Interface for Parallelizing Stochastic Algorithms

Stochastic algorithms are efficient approaches to solving machine learning and optimization problems. In this paper, we propose a general framework called Splash for parallelizing stochastic algorithms on multi-node distributed systems. Splash consists of a programming interface and an execution engine. Using the programming interface, the user develops sequential stochastic algorithms without concerning any detail about distributed computing. The algorithm is then automatically parallelized by a communication-efficient execution engine. We provide theoretical justifications on the optimal rate of convergence for parallelizing stochastic gradient descent. Splash is built on top of Apache Spark. The real-data experiments on logistic regression, collaborative filtering and topic modeling verify that Splash yields order-of-magnitude speedup over single-thread stochastic algorithms and over state-of-the-art implementations on Spark.

cs.LG↗

arXiv2015

Parallel Correlation Clustering on Big Graphs

Given a similarity graph between items, correlation clustering (CC) groups similar items together and dissimilar ones apart. One of the most popular CC algorithms is KwikCluster: an algorithm that serially clusters neighborhoods of vertices, and obtains a 3-approximation ratio. Unfortunately, KwikCluster in practice requires a large number of clustering rounds, a potential bottleneck for large graphs. We present C4 and ClusterWild!, two algorithms for parallel correlation clustering that run in a polylogarithmic number of rounds and achieve nearly linear speedups, provably. C4 uses concurrency control to enforce serializability of a parallel clustering process, and guarantees a 3-approximation ratio. ClusterWild! is a coordination free algorithm that abandons consistency for the benefit of better scaling; this leads to a provably small loss in the 3-approximation ratio. We provide extensive experimental results for both algorithms, where we outperform the state of the art, both in terms of clustering accuracy and running time. We show that our algorithms can cluster billion-edge graphs in under 5 seconds on 32 cores, while achieving a 15x speedup.

cs.DC↗

arXiv2015

Adding vs. Averaging in Distributed Primal-Dual Optimization

Distributed optimization methods for large-scale machine learning suffer from a communication bottleneck. It is difficult to reduce this bottleneck while still efficiently and accurately aggregating partial work from different machines. In this paper, we present a novel generalization of the recent communication-efficient primal-dual framework (CoCoA) for distributed optimization. Our framework, CoCoA+, allows for additive combination of local updates to the global parameters at each iteration, whereas previous schemes with convergence guarantees only allow conservative averaging. We give stronger (primal-dual) convergence rate guarantees for both CoCoA as well as our new variants, and generalize the theory for both methods to cover non-smooth convex loss functions. We provide an extensive experimental comparison that shows the markedly improved performance of CoCoA+ on several real-world distributed datasets, especially when scaling up the number of machines.

cs.LG↗

arXiv2015

On the accuracy of self-normalized log-linear models

Calculation of the log-normalizer is a major computational obstacle in applications of log-linear models with large output spaces. The problem of fast normalizer computation has therefore attracted significant attention in the theoretical and applied machine learning literature. In this paper, we analyze a recently proposed technique known as "self-normalization", which introduces a regularization term in training to penalize log normalizers for deviating from zero. This makes it possible to use unnormalized model scores as approximate probabilities. Empirical evidence suggests that self-normalization is extremely effective, but a theoretical understanding of why it should work, and how generally it can be applied, is largely lacking. We prove generalization bounds on the estimated variance of normalizers and upper bounds on the loss in accuracy due to self-normalization, describe classes of input distributions that self-normalize easily, and construct explicit examples of high-variance input distributions. Our theoretical results make predictions about the difficulty of fitting self-normalized models to several classes of distributions, and we conclude with empirical validation of these predictions.

stat.ML↗

arXiv2015

Variational consensus Monte Carlo

Practitioners of Bayesian statistics have long depended on Markov chain Monte Carlo (MCMC) to obtain samples from intractable posterior distributions. Unfortunately, MCMC algorithms are typically serial, and do not scale to the large datasets typical of modern machine learning. The recently proposed consensus Monte Carlo algorithm removes this limitation by partitioning the data and drawing samples conditional on each partition in parallel (Scott et al, 2013). A fixed aggregation function then combines these samples, yielding approximate posterior samples. We introduce variational consensus Monte Carlo (VCMC), a variational Bayes algorithm that optimizes over aggregation functions to obtain samples from a distribution that better approximates the target. The resulting objective contains an intractable entropy term; we therefore derive a relaxation of the objective and show that the relaxed problem is blockwise concave under mild conditions. We illustrate the advantages of our algorithm on three inference tasks from the literature, demonstrating both the superior quality of the posterior approximation and the moderate overhead of the optimization step. Our algorithm achieves a relative error reduction (measured against serial MCMC) of up to 39% compared to consensus Monte Carlo on the task of estimating 300-dimensional probit regression parameter expectations; similarly, it achieves an error reduction of 92% on the task of estimating cluster comembership probabilities in a Gaussian mixture model with 8 components in 8 dimensions. Furthermore, these gains come at moderate cost compared to the runtime of serial MCMC, achieving near-ideal speedup in some instances.

stat.ML↗

arXiv2015

On the Computational Complexity of High-Dimensional Bayesian Variable Selection

We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee both variable-selection consistency and rapid mixing of a particular Metropolis-Hastings algorithm. The mixing time is linear in the number of covariates up to a logarithmic factor. Our proof controls the spectral gap of the Markov chain by constructing a canonical path ensemble that is inspired by the steps taken by greedy algorithms for variable selection.

math.ST↗

arXiv2015

Learning Transferable Features with Deep Adaptation Networks

Recent studies reveal that a deep neural network can learn transferable features which generalize well to novel tasks for domain adaptation. However, as deep features eventually transition from general to specific along the network, the feature transferability drops significantly in higher layers with increasing domain discrepancy. Hence, it is important to formally reduce the dataset bias and enhance the transferability in task-specific layers. In this paper, we propose a new Deep Adaptation Network (DAN) architecture, which generalizes deep convolutional neural network to the domain adaptation scenario. In DAN, hidden representations of all task-specific layers are embedded in a reproducing kernel Hilbert space where the mean embeddings of different domain distributions can be explicitly matched. The domain discrepancy is further reduced using an optimal multi-kernel selection method for mean embedding matching. DAN can learn transferable features with statistical guarantees, and can scale linearly by unbiased estimate of kernel embedding. Extensive empirical evidence shows that the proposed architecture yields state-of-the-art image classification error rates on standard domain adaptation benchmarks.

cs.LG↗

arXiv2015

A General Analysis of the Convergence of ADMM

We provide a new proof of the linear convergence of the alternating direction method of multipliers (ADMM) when one of the objective terms is strongly convex. Our proof is based on a framework for analyzing optimization algorithms introduced in Lessard et al. (2014), reducing algorithm convergence to verifying the stability of a dynamical system. This approach generalizes a number of existing results and obviates any assumptions about specific choices of algorithm parameters. On a numerical example, we demonstrate that minimizing the derived bound on the convergence rate provides a practical approach to selecting algorithm parameters for particular ADMM instances. We complement our upper bound by constructing a nearly-matching lower bound on the worst-case rate of convergence.

math.OC↗

arXiv2015

TuPAQ: An Efficient Planner for Large-scale Predictive Analytic Queries

The proliferation of massive datasets combined with the development of sophisticated analytical techniques have enabled a wide variety of novel applications such as improved product recommendations, automatic image tagging, and improved speech-driven interfaces. These and many other applications can be supported by Predictive Analytic Queries (PAQs). A major obstacle to supporting PAQs is the challenging and expensive process of identifying and training an appropriate predictive model. Recent efforts aiming to automate this process have focused on single node implementations and have assumed that model training itself is a black box, thus limiting the effectiveness of such approaches on large-scale problems. In this work, we build upon these recent efforts and propose an integrated PAQ planning architecture that combines advanced model search techniques, bandit resource allocation via runtime algorithm introspection, and physical optimization via batching. The result is TuPAQ, a component of the MLbase system, which solves the PAQ planning problem with comparable quality to exhaustive strategies but an order of magnitude more efficiently than the standard baseline approach, and can scale to models trained on terabytes of data across hundreds of machines.

cs.DB↗

arXiv2015

Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds

We study the following generalized matrix rank estimation problem: given an $n \times n$ matrix and a constant $c \geq 0$, estimate the number of eigenvalues that are greater than $c$. In the distributed setting, the matrix of interest is the sum of $m$ matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate $Ω(n^2)$ bits, which is order-equivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only $\widetilde O(n)$ bits. The upper bound is matched by an $Ω(n)$ lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments.

cs.DS↗

arXiv2014

Active Learning for Crowd-Sourced Databases

Crowd-sourcing has become a popular means of acquiring labeled data for a wide variety of tasks where humans are more accurate than computers, e.g., labeling images, matching objects, or analyzing sentiment. However, relying solely on the crowd is often impractical even for data sets with thousands of items, due to time and cost constraints of acquiring human input (which cost pennies and minutes per label). In this paper, we propose algorithms for integrating machine learning into crowd-sourced databases, with the goal of allowing crowd-sourcing applications to scale, i.e., to handle larger datasets at lower costs. The key observation is that, in many of the above tasks, humans and machine learning algorithms can be complementary, as humans are often more accurate but slow and expensive, while algorithms are usually less accurate, but faster and cheaper. Based on this observation, we present two new active learning algorithms to combine humans and algorithms together in a crowd-sourced database. Our algorithms are based on the theory of non-parametric bootstrap, which makes our results applicable to a broad class of machine learning models. Our results, on three real-life datasets collected with Amazon's Mechanical Turk, and on 15 well-known UCI data sets, show that our methods on average ask humans to label one to two orders of magnitude fewer items to achieve the same accuracy as a baseline that labels random images, and two to eight times fewer questions than previous active learning schemes.

cs.LG↗

arXiv2014

The Missing Piece in Complex Analytics: Low Latency, Scalable Model Management and Serving with Velox

To support complex data-intensive applications such as personalized recommendations, targeted advertising, and intelligent services, the data management community has focused heavily on the design of systems to support training complex models on large datasets. Unfortunately, the design of these systems largely ignores a critical component of the overall analytics process: the deployment and serving of models at scale. In this work, we present Velox, a new component of the Berkeley Data Analytics Stack. Velox is a data management system for facilitating the next steps in real-world, large-scale analytics pipelines: online model management, maintenance, and serving. Velox provides end-user applications and services with a low-latency, intuitive interface to models, transforming the raw statistical models currently trained using existing offline large-scale compute frameworks into full-blown, end-to-end data products capable of recommending products, targeting advertisements, and personalizing web content. To provide up-to-date results for these complex models, Velox also facilitates lightweight online model maintenance and selection (i.e., dynamic weighting). In this paper, we describe the challenges and architectural considerations required to achieve this functionality, including the abilities to span online and offline systems, to adaptively adjust model materialization strategies, and to exploit inherent statistical properties such as model error tolerance, all while operating at "Big Data" scale.

cs.DB↗

arXiv2014

Joint modeling of multiple time series via the beta process with application to motion capture segmentation

We propose a Bayesian nonparametric approach to the problem of jointly modeling multiple related time series. Our model discovers a latent set of dynamical behaviors shared among the sequences, and segments each time series into regions defined by a subset of these behaviors. Using a beta process prior, the size of the behavior set and the sharing pattern are both inferred from data. We develop Markov chain Monte Carlo (MCMC) methods based on the Indian buffet process representation of the predictive distribution of the beta process. Our MCMC inference algorithm efficiently adds and removes behaviors via novel split-merge moves as well as data-driven birth and death proposals, avoiding the need to consider a truncated model. We demonstrate promising results on unsupervised segmentation of human motion capture data.

stat.ME↗

arXiv2014

On the Convergence Rate of Decomposable Submodular Function Minimization

Submodular functions describe a variety of discrete problems in machine learning, signal processing, and computer vision. However, minimizing submodular functions poses a number of algorithmic challenges. Recent work introduced an easy-to-use, parallelizable algorithm for minimizing submodular functions that decompose as the sum of "simple" submodular functions. Empirically, this algorithm performs extremely well, but no theoretical analysis was given. In this paper, we show that the algorithm converges linearly, and we provide upper and lower bounds on the rate of convergence. Our proof relies on the geometry of submodular polyhedra and draws on results from spectral graph theory.

math.OC↗

arXiv2014

Spectral Methods meet EM: A Provably Optimal Algorithm for Crowdsourcing

Crowdsourcing is a popular paradigm for effectively collecting labels at low cost. The Dawid-Skene estimator has been widely used for inferring the true labels from the noisy labels provided by non-expert crowdsourcing workers. However, since the estimator maximizes a non-convex log-likelihood function, it is hard to theoretically justify its performance. In this paper, we propose a two-stage efficient algorithm for multi-class crowd labeling problems. The first stage uses the spectral method to obtain an initial estimate of parameters. Then the second stage refines the estimation by optimizing the objective function of the Dawid-Skene estimator via the EM algorithm. We show that our algorithm achieves the optimal convergence rate up to a logarithmic factor. We conduct extensive experiments on synthetic and real datasets. Experimental results demonstrate that the proposed algorithm is comparable to the most accurate empirical approach, while outperforming several other recently proposed methods.

stat.ML↗

arXiv2014

Communication-Efficient Distributed Dual Coordinate Ascent

Communication remains the most significant bottleneck in the performance of distributed optimization algorithms for large-scale machine learning. In this paper, we propose a communication-efficient framework, CoCoA, that uses local computation in a primal-dual setting to dramatically reduce the amount of necessary communication. We provide a strong convergence rate analysis for this class of algorithms, as well as experiments on real-world distributed datasets with implementations in Spark. In our experiments, we find that as compared to state-of-the-art mini-batch versions of SGD and SDCA algorithms, CoCoA converges to the same .001-accurate solution quality on average 25x as quickly.

cs.LG↗

arXiv2014

Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates

Working under a model of privacy in which data remains private even from the statistician, we study the tradeoff between privacy guarantees and the utility of the resulting statistical estimators. We prove bounds on information-theoretic quantities, including mutual information and Kullback-Leibler divergence, that depend on the privacy guarantees. When combined with standard minimax techniques, including the Le Cam, Fano, and Assouad methods, these inequalities allow for a precise characterization of statistical rates under local privacy constraints. We provide a treatment of several canonical families of problems: mean estimation, parameter estimation in fixed-design regression, multinomial probability estimation, and nonparametric density estimation. For all of these families, we provide lower and upper bounds that match up to constant factors, and exhibit new (optimal) privacy-preserving mechanisms and computationally efficient estimators that achieve the bounds.

math.ST↗